Hamas has largely stayed out of direct confrontation with Israel since a war more than three years ago although other factions have launched cross-border attacks.
Gaza militants fired more than 30 rockets and mortar shells on Saturday alone, bringing the week's rocket tally to more than 150, according to the Israeli military.
Over a million people in southern Israel live in range of the rockets and the Israeli military has advised the residents to make sure they can get to a bomb shelter quickly.
An Egyptian official in Ramallah said that he is trying to broker a cease-fire between the Palestinian militant groups and Israel. He said he hoped calm would be restored by late Saturday. Egypt traditionally plays a mediating role between the sides.
The official spoke on condition of anonymity because negotiations were ongoing. Israel had no immediate official comment on the Egyptian efforts, but Hamas official Ayman Taha said his group would cease fire if Israel does. Soon after Taher spoke more rockets fired from Gaza exploded in southern Israel. It was not clear which group fired them.
A previous cease-fire reached late Tuesday did not hold long.
The military chief of staff, Lt. Gen. Benny Gantz, convened "an urgent meeting" to discuss ways of dealing with a week of persistent Palestinian attacks, a spokesman said, while Defense Minister Ehud Barak consulted with senior officials Saturday evening. There was no official statement afterward but the Israeli media reported that the decision was made "to meet fire with fire and quiet with quiet."
Israel's military said it carried out a series of air strikes Saturday in response to the rockets. At least four militants were killed and over 20 Palestinians, both combatants and civilians, were injured, according to Gaza health official Ashraf Al Kedra.
Hamas' military wing, the Izzedine al-Qassam Brigades, vowed to escalate the fighting.
"If the last rounds were not enough to deliver the message we are ready to crush the enemy and to curb its arrogance and to respond to aggression in a strong way," the group said in a statement released Saturday afternoon.
A few hours later an Israeli air strike killed a Gaza militant riding a scooter. The Popular Resistance Committees, a Hamas-allied militant group that often fires rockets at Israel, said he was a member. Witnesses said the force of the explosion caused a wall of a nearby building to collapse, injuring several people. The Israeli military said they targeted the militant as he was preparing to launch rockets.
In an earlier attack the same day, an Israeli air strike killed a Palestinian militant as he rode his motorcycle in Gaza. Several buildings used by Hamas, which rules the coastal territory, were also damaged.
The Israeli military said it hit "terror sites" and that it holds Hamas accountable for any attacks originating from the Gaza Strip.
Also, an Israeli was seriously wounded when a Gaza rocket smashed through a factory roof. Rockets from Gaza also hit a school in the Israeli border town of Sderot. The school was closed for the weekend so no one was hurt but the building was damaged. Several other Israelis were injured in Palestinian rocket attacks during the week.
"We have been enduring years of relentless Palestinian escalations and this is just one of them," said David Buskila, mayor of the Israeli border town Sderot where the school was hit. "We trust the Israeli military and government to restore calm," he said.
The spike in violence began Monday when two gunmen infiltrated into Israel along its border with Egypt's Sinai peninsula and killed an Israeli Arab construction worker who was part of a crew building a border security fence meant to avert such attacks. Later, a little known al-Qaeda-inspired group, the Mujahedeen Shura Council of Jerusalem, claimed responsibility for the attack and identified the gunmen as an Egyptian and a Saudi.
Since then, Gaza gunmen have fired barrages of rockets and mortar rounds at Israel, wounding several Israelis and damaging apartment buildings. Israel has carried out air strikes on militant targets that killed at least 12 Palestinians and wounded dozens so far.